Consider a Galton-Watson process, $W_0$, $W_1$, $W_2$ $\ldots$, where $W_0=1$ and the next random variables are defined by the following recursion, $$ W_t = \sum\limits_{i=0}^{W_{t-1}} \xi_i, $$ where $\xi_i$ are independent and identical non negative random variables. It is well known that if $E[\xi]<1$, then the process is subcritical, meaning that there exists a $C < 1 $ such that $P(\tau > k) \leq C^k$, where $\tau = \inf \{t \, : \, W_t = \emptyset \}$. This means that the process dies out exponentially fast.
Where can I find a proof that under the same hypothesis there exists a $C<1$ such that $$ P ( \sum\limits_{i=0}^{\tau} W_i > k ) < C^k, $$ i.e., not only the survival time $\tau$ decays exponentially fast but also the total number of individual of the process?
Let $\phi(\lambda)=\ln\mathbb{E}[e^{\lambda \xi_1}]$ assuming it exists for $\lambda >0$
And $\phi_n(\lambda)$ defined as $\phi_{1}(\lambda)=\phi(\lambda)$ and $\phi_{n+1}(\lambda)=\phi(\lambda + \phi_{n}(\lambda))$
Then:
$$\mathbb{E}\left[e^{\lambda \sum_{i=1}^nW_i}\right]=e^{\phi_n(\lambda)}$$
thus,
$$\mathbb{E}\left[e^{\lambda \sum_{i=1}^{\infty}W_i}\right]=e^{\phi_{\infty}(\lambda)}$$ where $\phi_{\infty}(\lambda)=\phi(\lambda + \phi_{\infty}(\lambda))$.
because $\phi'(0)=\mathbb{E}[\xi_1]<1$ for $\lambda>\lambda' >0$ small enough, $\phi_{\infty}(\lambda')$ will be defined and thus, you have what you want by Markov's inequality:
$$\mathbb{P}\left(\sum_{i=1}^\infty W_i>k\right)=\mathbb{P}\left(e^{\lambda'\sum_{i=1}^\infty W_i}\geq e^{\lambda'k}\right)\leq e^{\phi_{\infty}(\lambda')}e^{-\lambda' k}$$